Strongly Confined CsPbBr3 Quantum Dots as Quantum Emitters and Building Blocks for Rhombic Superlattices

The success of the colloidal semiconductor quantum dots (QDs) field is rooted in the precise synthetic control of QD size, shape, and composition, enabling electronically well-defined functional nanomaterials that foster fundamental science and motivate diverse fields of applications. While the exploitation of the strong confinement regime has been driving commercial and scientific interest in InP or CdSe QDs, such a regime has still not been thoroughly explored and exploited for lead-halide perovskite QDs, mainly due to a so far insufficient chemical stability and size monodispersity of perovskite QDs smaller than about 7 nm. Here, we demonstrate chemically stable strongly confined 5 nm CsPbBr3 colloidal QDs via a postsynthetic treatment employing didodecyldimethylammonium bromide ligands. The achieved high size monodispersity (7.5% ± 2.0%) and shape-uniformity enables the self-assembly of QD superlattices with exceptional long-range order, uniform thickness, an unusual rhombic packing with an obtuse angle of 104°, and narrow-band cyan emission. The enhanced chemical stability indicates the promise of strongly confined perovskite QDs for solution-processed single-photon sources, with single QDs showcasing a high single-photon purity of 73% and minimal blinking (78% “on” fraction), both at room temperature.


SAXS model fitting
The scattering pattern was fitted using the analytical model used in F. Krieg et al. 11 In short, we describe the model scattering intensity as: The fitting parameters are: ( ) the first (smallest) NC side length, ( ) the aspect ratio of the second to first NC side length, ( ) the aspect ratio of the third to first NC side length, ( ) the standard-deviation of the size-distribution (relative) and ( 0 ) the forward scattering intensity scalar. 1 The effect of polydispersity is taken into account by performing the numerical average (summation) over the Gaussian distribution ( , , ), here consisting of = 51 discrete points, which is discretized over the range (1 − 3 ) < < (1 + 3 ) and is weighted by volume = 3 • • . The formfactor contribution stemming from the NC shape � , , , � is calculated using a theoretical orthorhombic model according to literature 2 such that ( = 0) = 1, where denotes the shortest NC side length corresponding to the respective size-distribution weight such that the other sides are scaled by and .
The scattering pattern was fitted over the angular regime of 0.11 < < 6.2 nm -1 , as shown in Figure 1b of the main text. During the fitting process, all parameters were optimized (see Table  S1 for results) -only the background ( ) was kept fixed to mitigate potential cross-corelation uncertainties with other fitting variables. Overall, we find outstanding agreement of experimental and model scattering patterns, as quantitatively evidenced by the goodnesss-of-fit value 2 = 1.94.

The Debye scattering equation (DSE) method
The DSE, employed for fitting the WAXTS data reported in the main text, provides the average differential elastic cross-section (or the powder diffraction pattern) of a randomly oriented powder from the distribution of interatomic distances between atomic pairs, without any assumption of periodicity and order: 3,4 S3 where Q = 4πsinθ/λ is the magnitude of the scattering vector, λ is the radiation wavelength, is the atomic form factor of element , is the interatomic distance between atoms and , is the total number of atoms and and are the thermal atomic displacement parameter and the site occupancy factor associated to each atomic species, respectively. The first summation in the above equation includes the contributions of zero distances between one atom and itself and the second term (the interference term) the non-zero interatomic distances = � − �.
To compute the DSE according to the Debussy 5 strategy, a bivariate population of atomistic models of CsPbBr3 QDs was generated, by packing the orthorhombic building block (in the Pnma setting) along three growth directions: one (independent) parallel to the b axis of the unit cell and the other two (characterized by an equal number of steps) obtained as a combination of a and c axes (a+c and a-c).
This model construction was selected to obtain a pseudocubic morphology for CsPbBr3 QDs, exposing four {101} and two {010} facets (indices in the orthorhombic Pnma setting), in agreement with HRTEM observations, as detailed in the main text and in ref. 6. We note here that two {010} facets, i.e. 010 and 0-10, make the so called pinacoid, while the remaining {101} faces are the four equivalent lateral faces of a rhombic prism aligned in the b unit cell direction.
All six facets were terminated with CsBr moieties, the site occupancy of which are subsequently refined to adjust the QD surface termination against the experimental data.
The step between two consecutive clusters in the population, along each growth direction, was set to 5.88 Å (i.e. equivalent to the edge of a primitive cubic unit cell).
To improve the quality of the fit and to account for the apparent "cubicity" of the experimental WAXTS pattern (i.e. with practically absent orthorhombic superstructure peaks), slip planes were added to the QD models (one per each cluster of the population) according to the rule ½<101>(010) (in Pnma setting), as detailed in ref. 7.
The interatomic distances of these clusters are then computed, sampled (according to a Gaussian sampling strategy, to reduce the computational times), 8 and used to compute the model DSE.

Photoluminescence quantum yield
The photoluminescence quantum yield of our 5 nm CsPbBr3 QDs is about 65% in colloidal dispersion and above 50% in film.

Additional AFM images
Typical superlattices (SL) of 5 nm CsPbBr3 quantum dots (QDs) exhibit thicknesses of several hundreds of nanometers, see Figure S1 for several representative SLs. The SL indicated by the cyan rectangle in Figure S1b corresponds to the SL shown in Figure 2c of the main text. This SL exhibits a thickness of about 0.7 µm and a root-mean-square roughness of 11.86 nm. The indicated "diagonal" distance between QDs in the SL orientation [1][2][3][4][5][6][7][8][9][10] in our model (12.9 nm) is in good agreement with the experimentally determined distance (12.3 nm) in (d).

Image recognition methodology for optical microscopy images
We start by converting an input image from an optical microscope into a gray-scale image, with a single channel containing the light intensity. After a pre-processing step to sharpen the image utilizing the open-source computer vision library OpenCV, 9 edge detection is performed via thresholding. A fixed-level threshold ultimately converts the image into a binary one. Here, we would like to note that while the currently applied threshold level was arbitrary, and thus subject to some user bias, future implementation could reduce statistical bias via a suitable descriptor, e.g. based on particle concentration. Figure S6 shows a comparison of the image before and after processing.

Original image
Image after sharpening Image after thresholding Figure S6 | Sequential steps in the image recognition algorithm.
Next, we utilize the OpenCV library 9 to perform a two-step contour search on the binary image, as an essential tool for shape analysis and object detection and recognition. After an initial contour search, obvious failures are rejected based on considerations of the expected minimum and maximum size of a detected object. Each remaining detected object is approximated by a polygonal curve consisting of four segments. Using basic linear algebra, we obtain the norms of the detected vectors and the enclosed angles. As a final result, we plot the histogram of all angles (four per detected object), as shown in the inset of Figure 2a in the main text, also reproduced here as Figure S7. Clearly, the detected angles suggest that QD SLs exhibit a non-orthogonal shape, with a characteristic obtuse angle of about 104⁰.

Average QD size derived from DSE fits of the Porod small-angle region and WAXTS data
The simultaneous DSE fitting of the Porod small-angle region and WAXTS data displayed in Figure 1d of the main text yields an oblate shape with average edge lengths of 4.5 nm, 5.2 nm, and 5.2 nm, respectively, and size dispersions (i.e. standard deviations of the bivariate lognormal distribution) of 17.2%, 7.8%, and 7.8%, respectively, corroborating the SAXS-derived values displayed in Figure 1b of the main text. The edge lengths derived and the standard deviations from the DSE fitting are mass-based average values.

GISAXS -the importance of accounting for the form factor
In a small X-Ray scattering experiment of particles with both positional and orientational order, the detected scattering intensity ⃗ � ��⃗ � can be written as where ��⃗ denotes the scattering vector, � �⃗ ( ��⃗ ) denotes the particle form-factor and � �⃗ ( ��⃗ ) denotes the structure factor -note that all variables are given as vectors in 3D reciprocal space. In the following, we address the relevant aspects causing the observed modulation of Braggpeak intensities, as shown in Figure 2i of the main text.
The scattering vector ��⃗ is determined by the experimental geometry, so the X-Ray incidence angle as well as the scattering angle with respect to the sample orientation. The particle formfactor � �⃗ ( ��⃗ ) is governed by the QD shape. As an example, the solution SAXS pattern of the diluted QDs presented in Figure 1b of the main text shows the spherically averaged form factor ( ) = 〈 � �⃗ ( ��⃗ )〉. The structure factor � �⃗ ( ��⃗ ) is governed by the relative arrangements of individual scattering entities (atoms for GIWAXS and QDs for GISAXS), which can be defined, in the simplest cases, by the lattice periodicity only, or, more often, by the relative location of the scatterers within a unit cell. Hence, � �⃗ (� �⃗) contains direction-dependent S11 information on the SL symmetry (crystal system and space group), periodicity (lattice parameter(s)), and "quality" (size, mosaicity, etc.).
As it is evident from the equation above, if a scattering intensity is measured at a specific ��⃗ , both form factor and structure factor need to be non-zero. Vice versa, if the either form factor or structure factor is zero, then also the scattering intensity will be supressed.
An evident example of this phenomenon is found in the horizontal cuts of the GISAXS patterns shown in Figure 2j of the main text. Here, the dashed lines indicate Q positions where one would expect SL Bragg peaks thanks to the in-plane order, yet some are experimentally unobserved. By superimposing the theoretical form factor of the QDs (calculated, for simplicity, assuming monodispersity with QD rhombic edges normal to the SL axis cSL) it becomes evident that the experimentally missing Bragg reflections fall directly within the minima of the QD form factor. Specifically, this explains why Bragg peaks at about 0.12 Å -1 and 0.20 Å -1 are highly suppressed, while the Bragg peak in between, at 0.15 Å -1 , is strongly enhanced (as it is located near a strong form-factor maximum). A prominent example of this effect has been discussed by M.R. Jones et al. 10

GISAXS -discarded alternative structural models
As counterproof of the proposed C-centered orthorhombic cell of the SL (which, as discussed, is nothing else than a simple textbook transformation of the primitive monoclinic 8.2 x 8.2 x 8.2 nm, γ = 104° lattice), we also attempted to index the obtained GISAXS patterns using an analogous primitive orthogonal lattice, maintaining cell edges lengths and forcing γ to 90°. A comparison between the spot indexing obtained by primitive monoclinic and primitive orthogonal lattices (which, inter alia, possess very different molar volumes) is shown in Figure S10. As neither the real nor the reciprocal lattice coincide (or are at least commensurate), in Figure S10c a few intense spots remain unindexed (eliminating the overlay of theoretical peak positions, SL Bragg reflections are more clearly visible). Ab initio indexing of the all observed spots (performed by GIDInd, doi.org/10.1107/S1600576721006609), using a joint Qxy and Qz dataset (with Qz values taken from specular diffraction measurements), found a plausible c-unique primitive monoclinic lattice solution, nicely matching crystal morphology and leading to an overall orthorhombic lattice symmetry by simple axis transformation. Understandably, after transformation, the C-centered orthorhombic cell enables recovery of hkl triplets for all observed spots.

GISAXS of additional QD SLs
To provide evidence of reproducibility of the formation of the proposed C-centered orthorhombic superlattice, we used the equivalent primitive monoclinic setting to interpret independently measured GISAXS images, acquired on a second sample more than one year after the initial measurements (those presented in Figure 2 of the main text and in Figure S10 above). In this late experiment, the sample-detector distance was adjusted to ca. 1.8 m, whereas images were taken at an incidence angle of 0.23° (aligned using the specular reflection).
As shown in Figure S11, also the new 2D GISAXS pattern could be indexed using a primitive monoclinic lattice. Here, the retrieved lattice parameters are a = b = 8.2 nm, c = 7.8 nm (with the c axis slightly reduced likely due to evaporation-induced compression effects, as observed in F. Krieg et al. 11 ), and γ = 102°. This indexing further provides compelling evidence that the C-centered orthorhombic symmetry is maintained, as it only depends on the a = b relationship, and not on the c axis length or γ angle value. Overall, the agreement between these two independently obtained QDs, SLs, and GISAXS measurements evidences the outstanding repeatability of QD synthesis, SL growth, measurement stability, and lattice indexing. Here, we assess the electronic structure of strongly confined 5 nm CsPbBr3 from a computational point of view. Figure S12a reproduces the atomistic model utilized in the DSE fitting of the WAXTS data, see Figure 1e of the main text. Figure S12b shows the associated projected density of states (PDOS) calculated at the DFT/PBE level of theory using the CP2K software package. The clean bandgap, void of trap states, is consistent the high PL quantum yield achieved in the experiment. The band edge states predominantly derive from Pb and Br atomic orbitals, as typical also for larger CsPbBr3 nanocrystals and bulk.

Control of the QD packing via the ligand length
In order to explore the effect of the capping ligand length on the self-assembly of 5 nm CsPbBr3 QDs, we replace the DDAB ligands (didodecyldimethylammonium bromide, (C12H25)2(CH3)2NBr, with sidechains comprising 12 carbons) with other quaternary ammonium salts of either shorter or longer hydrocarbon sidechains, specifically:
In the case of the long (C18H37)2(CH3)2NBr ligands, the larger softness value L/R leads to a "rounding" of the cuboidal shape of the 5 nm CsPbBr3 NCs. Self-assembly of such QDs results in the body-centered cubic type SL shown in Figure S13b, a SL often observed for nearly spherical NCs with a significant soft character (see e.g. J. Phys. Chem. Lett. 2015, 6, 13, 2406-2412and Nature Materials 2016. In contrast, the installation of shorter (C8H17)2(CH3)2NBr capping molecules on the surface of 5 nm CsPbBr3 NCs yields simple cubic SLs (see Figure S13a), due to the decreased softness L/R and, concomitantly, well-defined edges and corners of the cuboidal building blocks. Such observations support our hypothesis that the large volume fraction of soft DDAB ligands on the surface of our 5 nm CsPbBr3 nanocubes is key to explaining their atypical self-assembly into SLs of rhombic shape. More generally, engineering the softness of NCs via their ligand fraction is one way to direct the SL self-assembly into various lattice structures. In a recent study, see Nature Communications 2022, 13, 2587, we have experimentally and computationally addressed the magnitude and origin of size-dependent homogeneous (thermal) broadening in CsPbBr3 QDs, highlighting the pivotal role of the dynamic QD surface in phonon-induced emission broadening. In line with the findings of this prior work, we find here particularly pronounced PL broadening in small 5 nm CsPbBr3 QDs with large surface-tovolume ratios. As shown in Figure S14, the PL linewidth (full width at half maximum, fwhm, obtained via Lorentzian fitting) of single 5 nm QDs increases strongly from 2 meV at 4 K to 5 meV at 30 K, 15 meV at 80 K, and 85 meV at 300 K, respectively. In compiling the temperature dependence shown in Figure S14d, spanning from 4 K up to 300 K, we took care to select representative QDs in terms of their broadening. Following a single QD across the entire temperature range was not possible, due to the still limited optical stability of such small QDs.

Temperature-dependent blinking characteristics in single 5 nm CsPbBr 3 QDs
Figure S15 | Temperature-dependent blinking characteristics in single 5 nm CsPbBr 3 QDs. PL spectra were recorded at 4.6 K (a), 30 K (b), and 80 K (c), respectively, over the course of 60 s, with a 1-s integration time. Associated blinking traces at 4.6 K (d), 30 K (e), and 80 K (f), respectively, have been obtained from (a)-(c) via spectral integration across the PL peak, after background subtraction.
Commonly, cooling down QDs can suppress PL intensity blinking, as e.g. reported for II-VI QDs (see e.g. Phys. Rev. B 2001, 63 205316) and larger CsPbBr3 QDs (see e.g. ACS Nano 2016, 10,2,[2485][2486][2487][2488][2489][2490]. However, PL blinking in our 5 nm QDs still persists at cryogenic temperatures, as clearly observed when acquiring consecutive PL spectra. Suppressing PL S17 blinking at cryogenic temperature still requires further work on the surface passivation and optimization of sample preparation routines. Indeed, strong dilution factors, needed to prepare sparse QD films, are possibly the cause of a non-optimal ligand surface coverage in single-QD samples which could be responsible for the observed blinking phenomenon.